Numerical Linear Algebra

Biostat/Biomath M257

System information (for reproducibility):

versioninfo()

Julia Version 1.11.4
Commit 8561cc3d68d (2025-03-10 11:36 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: macOS (arm64-apple-darwin24.0.0)
  CPU: 12 × Apple M2 Max
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, apple-m2)
Threads: 8 default, 0 interactive, 4 GC (on 8 virtual cores)
Environment:
  JULIA_NUM_THREADS = 8
  JULIA_EDITOR = code

Load packages:

using Pkg

Pkg.activate(pwd())
Pkg.instantiate()
Pkg.status()

  Activating project at `~/Documents/github.com/ucla-biostat-257/2025spring/slides/08-numalgintro`

Status `~/Documents/github.com/ucla-biostat-257/2025spring/slides/08-numalgintro/Project.toml`
  [6e4b80f9] BenchmarkTools v1.6.0
  [0e44f5e4] Hwloc v3.3.0
  [bdcacae8] LoopVectorization v0.12.172
  [6f49c342] RCall v0.14.6
  [37e2e46d] LinearAlgebra v1.11.0
  [9a3f8284] Random v1.11.0

1 Introduction

• Topics in numerical algebra:

  • BLAS
    
  • solve linear equations \(\mathbf{A} \mathbf{x} = \mathbf{b}\)
  • regression computations \(\mathbf{X}^T \mathbf{X} \beta = \mathbf{X}^T \mathbf{y}\)
    
  • eigen-problems \(\mathbf{A} \mathbf{x} = \lambda \mathbf{x}\)
    
  • generalized eigen-problems \(\mathbf{A} \mathbf{x} = \lambda \mathbf{B} \mathbf{x}\)
    
  • singular value decompositions \(\mathbf{A} = \mathbf{U} \Sigma \mathbf{V}^T\)
    
  • iterative methods for numerical linear algebra

• Except for the iterative methods, most of these numerical linear algebra tasks are implemented in the BLAS and LAPACK libraries. They form the building blocks of most statistical computing tasks (optimization, MCMC).

• Our major goal (or learning objectives) is to

  1. know the complexity (flop count) of each task
  2. be familiar with the BLAS and LAPACK functions (what they do)
     
  3. do not re-invent wheels by implementing these dense linear algebra subroutines by yourself
     
  4. understand the need for iterative methods
     
  5. apply appropriate numerical algebra tools to various statistical problems

• All high-level languages (Julia, Matlab, Python, R) call BLAS and LAPACK for numerical linear algebra.

  • Julia offers more flexibility by exposing interfaces to many BLAS/LAPACK subroutines directly. See documentation.

2 BLAS

• BLAS stands for basic linear algebra subprograms.

• See netlib for a complete list of standardized BLAS functions.

• There are many implementations of BLAS.

  • Netlib provides a reference implementation.
    
  • Matlab uses Intel’s MKL (mathematical kernel libaries). MKL implementation is the gold standard on market. It is not open source but the compiled library is free for Linux and MacOS. However, not surprisingly, it only works on Intel CPUs.
    
  • Julia uses OpenBLAS. OpenBLAS is the best cross-platform, open source implementation. With the MKL.jl package, it’s also very easy to use MKL in Julia.

• There are 3 levels of BLAS functions.

  • Level 1: vector-vector operation
  • Level 2: matrix-vector operation
  • Level 3: matrix-matrix operation

Level	Example Operation	Name	Dimension	Flops
1	\(\alpha \gets \mathbf{x}^T \mathbf{y}\)	dot product	\(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\)	\(2n\)
1	\(\mathbf{y} \gets \mathbf{y} + \alpha \mathbf{x}\)	axpy	\(\alpha \in \mathbb{R}\), \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\)	\(2n\)
2	\(\mathbf{y} \gets \mathbf{y} + \mathbf{A} \mathbf{x}\)	gaxpy	\(\mathbf{A} \in \mathbb{R}^{m \times n}\), \(\mathbf{x} \in \mathbb{R}^n\), \(\mathbf{y} \in \mathbb{R}^m\)	\(2mn\)
2	\(\mathbf{A} \gets \mathbf{A} + \mathbf{y} \mathbf{x}^T\)	rank one update	\(\mathbf{A} \in \mathbb{R}^{m \times n}\), \(\mathbf{x} \in \mathbb{R}^n\), \(\mathbf{y} \in \mathbb{R}^m\)	\(2mn\)
3	\(\mathbf{C} \gets \mathbf{C} + \mathbf{A} \mathbf{B}\)	matrix multiplication	\(\mathbf{A} \in \mathbb{R}^{m \times p}\), \(\mathbf{B} \in \mathbb{R}^{p \times n}\), \(\mathbf{C} \in \mathbb{R}^{m \times n}\)	\(2mnp\)

• Typical BLAS functions support single precision (S), double precision (D), complex (C), and double complex (Z).

3 Examples

The form of a mathematical expression and the way the expression should be evaluated in actual practice may be quite different.

Some operations appear as level-3 but indeed are level-2.

Example 1. A common operation in statistics is column scaling or row scaling \[ \begin{eqnarray*} \mathbf{A} &=& \mathbf{A} \mathbf{D} \quad \text{(column scaling)} \\ \mathbf{A} &=& \mathbf{D} \mathbf{A} \quad \text{(row scaling)}, \end{eqnarray*} \] where \(\mathbf{D}\) is diagonal. For example, in generalized linear models (GLMs), the Fisher information matrix takes the form
\[ \mathbf{X}^T \mathbf{W} \mathbf{X}, \] where \(\mathbf{W}\) is a diagonal matrix with observation weights on diagonal.

Column and row scalings are essentially level-2 operations!

using BenchmarkTools, LinearAlgebra, Random

Random.seed!(257) # seed
n = 2000
A = rand(n, n) # n-by-n matrix
d = rand(n)  # n vector
D = Diagonal(d) # diagonal matrix with d as diagonal

2000×2000 Diagonal{Float64, Vector{Float64}}:
 0.0416032   ⋅         ⋅         ⋅       …   ⋅         ⋅         ⋅ 
  ⋅         0.887679   ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅        0.102233   ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅        0.08407      ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅       …   ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅       …   ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
 ⋮                                       ⋱                      
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅       …   ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅       …   ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅          0.213471   ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅        0.870533   ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅        0.318876

Dfull = diagm(d) # convert to full matrix

2000×2000 Matrix{Float64}:
 0.0416032  0.0       0.0       0.0      …  0.0       0.0       0.0
 0.0        0.887679  0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.102233  0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.08407     0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0      …  0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0      …  0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 ⋮                                       ⋱                      
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0      …  0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0      …  0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.213471  0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.870533  0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.318876

# this is calling BLAS routine for matrix multiplication: O(n^3) flops
# this is SLOW!
@benchmark $A * $Dfull

BenchmarkTools.Trial: 96 samples with 1 evaluation per sample.
 Range (min … max):  49.658 ms … 61.309 ms  ┊ GC (min … max): 0.00% … 1.87%
 Time  (median):     51.773 ms              ┊ GC (median):    2.24%
 Time  (mean ± σ):   52.085 ms ±  1.717 ms  ┊ GC (mean ± σ):  1.98% ± 1.11%
          ▂ ▂▃▄█▃▂▃                                            
  ▅▁▃▃▁▃█▅██████████▅▃▇▁▁▃▃▁▁▁▁▁▁▁▁▁▅▃▁▁▁▁▁▁▁▁▃▁▁▁▁▁▁▃▁▁▁▁▁▁▃ ▁
  49.7 ms         Histogram: frequency by time        59.1 ms <
 Memory estimate: 30.52 MiB, allocs estimate: 3.

# dispatch to special method for diagonal matrix multiplication.
# columnwise scaling: O(n^2) flops
@benchmark $A * $D

BenchmarkTools.Trial: 1649 samples with 1 evaluation per sample.
 Range (min … max):  967.375 μs …   6.854 ms  ┊ GC (min … max):  0.00% … 60.53%
 Time  (median):       2.989 ms               ┊ GC (median):     0.00%
 Time  (mean ± σ):     3.029 ms ± 606.120 μs  ┊ GC (mean ± σ):  17.42% ± 15.07%
                          ▁▂▁▁▇▆▂                ▃▅█▂▂           
  ▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▄███████▆▅▃▄▃▂▂▃▂▂▂▂▃▅▅▇█████▆▄▄▃▃▃▃▃▂ ▃
  967 μs           Histogram: frequency by time         4.18 ms <
 Memory estimate: 30.52 MiB, allocs estimate: 3.

# Or we can use broadcasting (with recycling)
@benchmark $A .* transpose($d)

BenchmarkTools.Trial: 1652 samples with 1 evaluation per sample.
 Range (min … max):  969.916 μs …   6.552 ms  ┊ GC (min … max):  0.00% … 61.18%
 Time  (median):       2.890 ms               ┊ GC (median):     0.00%
 Time  (mean ± σ):     3.023 ms ± 605.004 μs  ┊ GC (mean ± σ):  17.44% ± 15.11%
                          ▁▁▂▅█▆▂               ▄▅▅▆▂            
  ▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▄▆███████▅▄▄▃▃▃▂▂▁▂▂▃▅▆▆██████▄▄▃▃▃▂▂▃▂ ▄
  970 μs           Histogram: frequency by time         4.23 ms <
 Memory estimate: 30.52 MiB, allocs estimate: 3.

# in-place: avoid allocate space for result
# rmul!: compute matrix-matrix product A*B, overwriting A, and return the result.
@benchmark rmul!($A, $D)

BenchmarkTools.Trial: 8816 samples with 1 evaluation per sample.
 Range (min … max):  547.333 μs … 753.834 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     554.459 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   563.254 μs ±  23.063 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
  ▆█▇█▇▆▅▄▃▃▃▂▂▂▂▂▁▁▁▁▁▁▁▁▁▁  ▁                                 ▂
  █████████████████████████████████████████▇▇▇█▆▇▇▇▇▆▅▅▆▇▅▅▅▅▅▆ █
  547 μs        Histogram: log(frequency) by time        659 μs <
 Memory estimate: 0 bytes, allocs estimate: 0.

# In-place broadcasting 
@benchmark $A .= $A .* transpose($d)

BenchmarkTools.Trial: 3197 samples with 1 evaluation per sample.
 Range (min … max):  1.518 ms …  2.118 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     1.546 ms              ┊ GC (median):    0.00%
 Time  (mean ± σ):   1.561 ms ± 41.229 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
      ▂▅▅██▂                                                  
  ▁▂▂▅██████▇▆▄▄▃▃▃▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▁▂▁▂▁▁▁▂▁▁▁▁▁▁▁▁▁▁▁ ▂
  1.52 ms        Histogram: frequency by time        1.71 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.

Exercise: Try @turbo (SIMD) and @tturbo (multi-threaded SIMD) from LoopVectorization.jl package.

Note: In R or Matlab, diag(d) will create a full matrix. Be cautious using diag function: do we really need a full diagonal matrix?

using RCall

R"""
d <- runif(5)
diag(d)
"""

RObject{RealSxp}
          [,1]      [,2]     [,3]      [,4]      [,5]
[1,] 0.7934108 0.0000000 0.000000 0.0000000 0.0000000
[2,] 0.0000000 0.1197454 0.000000 0.0000000 0.0000000
[3,] 0.0000000 0.0000000 0.621123 0.0000000 0.0000000
[4,] 0.0000000 0.0000000 0.000000 0.6349068 0.0000000
[5,] 0.0000000 0.0000000 0.000000 0.0000000 0.4085076

This works only when Matlab is installed.

#| eval: false
using MATLAB

mat"""
d = rand(5, 1)
diag(d)

Example 2. Innter product between two matrices \(\mathbf{A}, \mathbf{B} \in \mathbb{R}^{m \times n}\) is often written as \[ \text{trace}(\mathbf{A}^T \mathbf{B}), \text{trace}(\mathbf{B} \mathbf{A}^T), \text{trace}(\mathbf{A} \mathbf{B}^T), \text{ or } \text{trace}(\mathbf{B}^T \mathbf{A}). \] They appear as level-3 operation (matrix multiplication with \(O(m^2n)\) or \(O(mn^2)\) flops).

Random.seed!(123)
n = 2000
A, B = randn(n, n), randn(n, n)

# slow way to evaluate tr(A'B): 2mn^2 flops
@benchmark tr(transpose($A) * $B)

BenchmarkTools.Trial: 95 samples with 1 evaluation per sample.
 Range (min … max):  49.499 ms … 67.595 ms  ┊ GC (min … max): 0.00% … 1.71%
 Time  (median):     52.340 ms              ┊ GC (median):    2.24%
 Time  (mean ± σ):   52.994 ms ±  2.647 ms  ┊ GC (mean ± σ):  2.03% ± 1.10%
     ▃        █▄                                               
  ▃▁▃█▅▃▆▁▁▅▃███▆███▇▁▅▇▅▅▅▃▅▁▁▁▅▃▃▁▃▃▁▅▁▃▁▁▅▃▁▁▁▁▁▁▁▃▁▁▁▁▁▁▅ ▁
  49.5 ms         Histogram: frequency by time        60.3 ms <
 Memory estimate: 30.52 MiB, allocs estimate: 3.

But \(\text{trace}(\mathbf{A}^T \mathbf{B}) = <\text{vec}(\mathbf{A}), \text{vec}(\mathbf{B})>\). The latter is level-1 BLAS operation with \(O(mn)\) flops.

# smarter way to evaluate tr(A'B): 2mn flops
@benchmark dot($A, $B)

BenchmarkTools.Trial: 10000 samples with 1 evaluation per sample.
 Range (min … max):  407.042 μs …   9.591 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     427.417 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   444.185 μs ± 205.582 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
    ▆█▆▄▂                                                        
  ▁▅██████▆▅▅▄▃▂▂▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▂
  407 μs           Histogram: frequency by time          638 μs <
 Memory estimate: 0 bytes, allocs estimate: 0.

Example 3. Similarly \(\text{diag}(\mathbf{A}^T \mathbf{B})\) can be calculated in \(O(mn)\) flops.

# slow way to evaluate diag(A'B): O(n^3)
@benchmark diag(transpose($A) * $B)

BenchmarkTools.Trial: 94 samples with 1 evaluation per sample.
 Range (min … max):  50.296 ms … 72.373 ms  ┊ GC (min … max): 0.00% … 1.71%
 Time  (median):     52.490 ms              ┊ GC (median):    2.29%
 Time  (mean ± σ):   53.271 ms ±  3.316 ms  ┊ GC (mean ± σ):  2.02% ± 1.09%
       ▅█                                                      
  ▄▄▆▃▅██▅▇▆▄▃▃▃▃▁▁▁▁▁▁▁▁▁▃▁▁▁▃▁▁▁▁▃▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃ ▁
  50.3 ms         Histogram: frequency by time        70.3 ms <
 Memory estimate: 30.53 MiB, allocs estimate: 6.

# smarter way to evaluate diag(A'B): O(n^2)
@benchmark Diagonal(vec(sum($A .* $B, dims = 1)))

BenchmarkTools.Trial: 1224 samples with 1 evaluation per sample.
 Range (min … max):  1.646 ms …   7.216 ms  ┊ GC (min … max):  0.00% … 54.60%
 Time  (median):     3.924 ms               ┊ GC (median):     0.00%
 Time  (mean ± σ):   4.081 ms ± 723.269 μs  ┊ GC (mean ± σ):  16.03% ± 14.14%
                          ▂▁█▇▃                ▁▇▅▃▁           
  ▂▁▁▁▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▄███████▄▃▃▃▃▃▂▂▃▁▁▁▃▅██████▇▃▃▃▂▃▃▃▂ ▃
  1.65 ms         Histogram: frequency by time         5.5 ms <
 Memory estimate: 30.53 MiB, allocs estimate: 7.

To get rid of allocation of intermediate arrays at all, we can just write a double loop or use dot function.

function diag_matmul!(d, A, B)
    m, n = size(A)
    @assert size(B) == (m, n) "A and B should have same size"
    fill!(d, 0)
    for j in 1:n, i in 1:m
        d[j] += A[i, j] * B[i, j]
    end
    Diagonal(d)
end

d = zeros(eltype(A), size(A, 2))
@benchmark diag_matmul!($d, $A, $B)

BenchmarkTools.Trial: 1469 samples with 1 evaluation per sample.
 Range (min … max):  3.339 ms …  3.613 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     3.397 ms              ┊ GC (median):    0.00%
 Time  (mean ± σ):   3.400 ms ± 55.130 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
  █                                                           
  █▇▅▅▅▄▄▄▃▄▃▃▃▃▄▄▄▅▅▅▅▄▄▄▄▄▄▃▃▃▃▃▃▂▃▂▃▂▂▂▃▃▂▂▃▃▂▂▂▂▂▂▂▂▂▂▁▂ ▃
  3.34 ms        Histogram: frequency by time        3.57 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.

Exercise: Try @turbo (SIMD) and @tturbo (multi-threaded SIMD) from LoopVectorization.jl package.

4 Memory hierarchy and level-3 fraction

Key to high performance is effective use of memory hierarchy. True on all architectures.

• Flop count is not the sole determinant of algorithm efficiency. Another important factor is data movement through the memory hierarchy.

Source: https://cs.brown.edu/courses/csci1310/2020/assign/labs/lab4.html

• In Julia, we can query the CPU topology by the Hwloc.jl package. For example, this laptop runs an Apple M2 Max chip with 4 efficiency cores and 8 performance cores.

using Hwloc

topology_graphical()

/------------------------------------------------------------------------------------------------------------------------------------------------------------------\
| Machine (2213MB total)                                                                                                                                           |
|                                                                                                                                                                  |
| /----------------------------------------------------------------------------------------------------------------------------------------\  /------------------\ |
| | Package L#0                                                                                                                            |  | CoProc opencl0d0 | |
| |                                                                                                                                        |  |                  | |
| | /------------------------------------------------------------------------------------------------------------------------------------\ |  | 38 compute units | |
| | | NUMANode L#0 P#0 (2213MB)                                                                                                          | |  |                  | |
| | \------------------------------------------------------------------------------------------------------------------------------------/ |  | 72 GB            | |
| |                                                                                                                                        |  \------------------/ |
| | /----------------------------------------------------------------\  /----------------------------------------------------------------\ |                       |
| | | L2 (4096KB)                                                    |  | L2 (16MB)                                                      | |                       |
| | \----------------------------------------------------------------/  \----------------------------------------------------------------/ |                       |
| |                                                                                                                                        |                       |
| | /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\ |                       |
| | | L1d (64KB)  |  | L1d (64KB)  |  | L1d (64KB)  |  | L1d (64KB)  |  | L1d (128KB) |  | L1d (128KB) |  | L1d (128KB) |  | L1d (128KB) | |                       |
| | \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/ |                       |
| |                                                                                                                                        |                       |
| | /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\ |                       |
| | | L1i (128KB) |  | L1i (128KB) |  | L1i (128KB) |  | L1i (128KB) |  | L1i (192KB) |  | L1i (192KB) |  | L1i (192KB) |  | L1i (192KB) | |                       |
| | \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/ |                       |
| |                                                                                                                                        |                       |
| | /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\ |                       |
| | | Core L#0    |  | Core L#1    |  | Core L#2    |  | Core L#3    |  | Core L#4    |  | Core L#5    |  | Core L#6    |  | Core L#7    | |                       |
| | |             |  |             |  |             |  |             |  |             |  |             |  |             |  |             | |                       |
| | | /---------\ |  | /---------\ |  | /---------\ |  | /---------\ |  | /---------\ |  | /---------\ |  | /---------\ |  | /---------\ | |                       |
| | | | PU L#0  | |  | | PU L#1  | |  | | PU L#2  | |  | | PU L#3  | |  | | PU L#4  | |  | | PU L#5  | |  | | PU L#6  | |  | | PU L#7  | | |                       |
| | | |         | |  | |         | |  | |         | |  | |         | |  | |         | |  | |         | |  | |         | |  | |         | | |                       |
| | | |   P#0   | |  | |   P#1   | |  | |   P#2   | |  | |   P#3   | |  | |   P#4   | |  | |   P#5   | |  | |   P#6   | |  | |   P#7   | | |                       |
| | | \---------/ |  | \---------/ |  | \---------/ |  | \---------/ |  | \---------/ |  | \---------/ |  | \---------/ |  | \---------/ | |                       |
| | \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/ |                       |
| |                                                                                                                                        |                       |
| | /----------------------------------------------------------------\                                                                     |                       |
| | | L2 (16MB)                                                      |                                                                     |                       |
| | \----------------------------------------------------------------/                                                                     |                       |
| |                                                                                                                                        |                       |
| | /-------------\  /-------------\  /-------------\  /-------------\                                                                     |                       |
| | | L1d (128KB) |  | L1d (128KB) |  | L1d (128KB) |  | L1d (128KB) |                                                                     |                       |
| | \-------------/  \-------------/  \-------------/  \-------------/                                                                     |                       |
| |                                                                                                                                        |                       |
| | /-------------\  /-------------\  /-------------\  /-------------\                                                                     |                       |
| | | L1i (192KB) |  | L1i (192KB) |  | L1i (192KB) |  | L1i (192KB) |                                                                     |                       |
| | \-------------/  \-------------/  \-------------/  \-------------/                                                                     |                       |
| |                                                                                                                                        |                       |
| | /-------------\  /-------------\  /-------------\  /-------------\                                                                     |                       |
| | | Core L#8    |  | Core L#9    |  | Core L#10   |  | Core L#11   |                                                                     |                       |
| | |             |  |             |  |             |  |             |                                                                     |                       |
| | | /---------\ |  | /---------\ |  | /---------\ |  | /---------\ |                                                                     |                       |
| | | | PU L#8  | |  | | PU L#9  | |  | | PU L#10 | |  | | PU L#11 | |                                                                     |                       |
| | | |         | |  | |         | |  | |         | |  | |         | |                                                                     |                       |
| | | |   P#8   | |  | |   P#9   | |  | |  P#10   | |  | |  P#11   | |                                                                     |                       |
| | | \---------/ |  | \---------/ |  | \---------/ |  | \---------/ |                                                                     |                       |
| | \-------------/  \-------------/  \-------------/  \-------------/                                                                     |                       |
| \----------------------------------------------------------------------------------------------------------------------------------------/                       |
\------------------------------------------------------------------------------------------------------------------------------------------------------------------/

• For example, Xeon X5650 CPU has a theoretical throughput of 128 DP GFLOPS but a max memory bandwidth of 32GB/s.

• Can we keep CPU cores busy with enough deliveries of matrix data and ship the results to memory fast enough to avoid backlog?
  Answer: use high-level BLAS as much as possible.

BLAS	Dimension	Mem. Refs.	Flops	Ratio
Level 1: \(\mathbf{y} \gets \mathbf{y} + \alpha \mathbf{x}\)	\(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\)	\(3n\)	\(2n\)	3:2
Level 2: \(\mathbf{y} \gets \mathbf{y} + \mathbf{A} \mathbf{x}\)	\(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\), \(\mathbf{A} \in \mathbb{R}^{n \times n}\)	\(n^2\)	\(2n^2\)	1:2
Level 3: \(\mathbf{C} \gets \mathbf{C} + \mathbf{A} \mathbf{B}\)	\(\mathbf{A}, \mathbf{B}, \mathbf{C} \in\mathbb{R}^{n \times n}\)	\(4n^2\)	\(2n^3\)	2:n

• Higher level BLAS (3 or 2) make more effective use of arithmetic logic units (ALU) by keeping them busy. Surface-to-volume effect.

Source: Jack Dongarra’s slides.

• A distinction between LAPACK and LINPACK (older version of R uses LINPACK) is that LAPACK makes use of higher level BLAS as much as possible (usually by smart partitioning) to increase the so-called level-3 fraction.

• To appreciate the efforts in an optimized BLAS implementation such as OpenBLAS (evolved from GotoBLAS), see the Quora question, especially the video. Bottomline is

Get familiar with (good implementations of) BLAS/LAPACK and use them as much as possible.

5 Effect of data layout

• Data layout in memory affects algorithmic efficiency too. It is much faster to move chunks of data in memory than retrieving/writing scattered data.

• Storage mode: column-major (Fortran, Matlab, R, Julia) vs row-major (C/C++).

• Cache line is the minimum amount of cache which can be loaded and stored to memory.

  • x86 CPUs: 64 bytes
    
  • ARM CPUs: 32 bytes

• In Julia, we can query the cache line size by Hwloc.jl.

# Apple Silicon (M1/M2 chips) don't have L3 cache
Hwloc.cachelinesize()

LoadError: Your system doesn't seem to have an L3 cache.
Your system doesn't seem to have an L3 cache.

Stacktrace:
 [1] cachelinesize()
   @ Hwloc ~/.julia/packages/Hwloc/IvkQ5/src/highlevel_api.jl:392
 [2] top-level scope
   @ In[62]:2

• Accessing column-major stored matrix by rows (\(ij\) looping) causes lots of cache misses.

• Take matrix multiplication as an example \[ \mathbf{C} \gets \mathbf{C} + \mathbf{A} \mathbf{B}, \quad \mathbf{A} \in \mathbb{R}^{m \times p}, \mathbf{B} \in \mathbb{R}^{p \times n}, \mathbf{C} \in \mathbb{R}^{m \times n}. \] Assume the storage is column-major, such as in Julia. There are 6 variants of the algorithms according to the order in the triple loops.

  • jki or kji looping:

# inner most loop
for i in 1:m
    C[i, j] = C[i, j] + A[i, k] * B[k, j]
end

- `ikj` or `kij` looping:

# inner most loop        
for j in 1:n
    C[i, j] = C[i, j] + A[i, k] * B[k, j]
end

• ijk or jik looping:

# inner most loop        
for k in 1:p
    C[i, j] = C[i, j] + A[i, k] * B[k, j]
end

• We pay attention to the innermost loop, where the vector calculation occurs. The associated stride when accessing the three matrices in memory (assuming column-major storage) is

Variant	A Stride	B Stride	C Stride
\(jki\) or \(kji\)	Unit	0	Unit
\(ikj\) or \(kij\)	0	Non-Unit	Non-Unit
\(ijk\) or \(jik\)	Non-Unit	Unit	0

Apparently the variants \(jki\) or \(kji\) are preferred.

"""
    matmul_by_loop!(A, B, C, order)

Overwrite `C` by `A * B`. `order` indicates the looping order for triple loop.
"""
function matmul_by_loop!(A::Matrix, B::Matrix, C::Matrix, order::String)
    
    m = size(A, 1)
    p = size(A, 2)
    n = size(B, 2)
    fill!(C, 0)
    
    if order == "jki"
        @inbounds for j = 1:n, k = 1:p, i = 1:m
            C[i, j] += A[i, k] * B[k, j]
        end
    end

    if order == "kji"
        @inbounds for k = 1:p, j = 1:n, i = 1:m
            C[i, j] += A[i, k] * B[k, j]
        end
    end
    
    if order == "ikj"
        @inbounds for i = 1:m, k = 1:p, j = 1:n
            C[i, j] += A[i, k] * B[k, j]
        end
    end

    if order == "kij"
        @inbounds for k = 1:p, i = 1:m, j = 1:n
            C[i, j] += A[i, k] * B[k, j]
        end
    end
    
    if order == "ijk"
        @inbounds for i = 1:m, j = 1:n, k = 1:p
            C[i, j] += A[i, k] * B[k, j]
        end
    end
    
    if order == "jik"
        @inbounds for j = 1:n, i = 1:m, k = 1:p
            C[i, j] += A[i, k] * B[k, j]
        end
    end
    
end

using Random

Random.seed!(123)
m, p, n = 2000, 100, 2000
A = rand(m, p)
B = rand(p, n)
C = zeros(m, n);

• \(jki\) and \(kji\) looping:

using BenchmarkTools

@benchmark matmul_by_loop!($A, $B, $C, "jki")

BenchmarkTools.Trial: 87 samples with 1 evaluation per sample.
 Range (min … max):  57.160 ms …  59.892 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     57.806 ms               ┊ GC (median):    0.00%
 Time  (mean ± σ):   57.879 ms ± 484.503 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
         ██ ▃▄  ▁▁ ▆ ▁▆▁                                        
  ▆▄▁▁▁▁▇██▆██▄▆██▇█▆███▆▇▄▄▁▁▁▄▁▁▁▁▁▁▁▁▁▄▁▄▁▁▁▁▁▄▁▁▁▁▄▁▁▁▁▁▁▄ ▁
  57.2 ms         Histogram: frequency by time         59.7 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.

@benchmark matmul_by_loop!($A, $B, $C, "kji")

BenchmarkTools.Trial: 27 samples with 1 evaluation per sample.
 Range (min … max):  185.930 ms … 190.930 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     187.927 ms               ┊ GC (median):    0.00%
 Time  (mean ± σ):   188.021 ms ±   1.026 ms  ┊ GC (mean ± σ):  0.00% ± 0.00%
                      █ █   ▃    ▃                               
  ▇▁▁▁▁▇▁▁▁▁▁▁▇▇▇▁▁▇▁▁█▇█▁▇▇█▇▇▁▇█▁▁▇▇▁▁▁▁▇▁▁▁▁▁▁▁▁▁▇▁▁▁▁▁▁▁▁▁▇ ▁
  186 ms           Histogram: frequency by time          191 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.

• \(ikj\) and \(kij\) looping:

@benchmark matmul_by_loop!($A, $B, $C, "ikj")

BenchmarkTools.Trial: 10 samples with 1 evaluation per sample.
 Range (min … max):  511.926 ms … 546.203 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     516.175 ms               ┊ GC (median):    0.00%
 Time  (mean ± σ):   518.729 ms ±   9.901 ms  ┊ GC (mean ± σ):  0.00% ± 0.00%
  ▁ ▁ █▁   ▁▁█                                                ▁  
  █▁█▁██▁▁▁███▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█ ▁
  512 ms           Histogram: frequency by time          546 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.

@benchmark matmul_by_loop!($A, $B, $C, "kij")

BenchmarkTools.Trial: 10 samples with 1 evaluation per sample.
 Range (min … max):  511.935 ms … 523.353 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     513.894 ms               ┊ GC (median):    0.00%
 Time  (mean ± σ):   516.421 ms ±   4.372 ms  ┊ GC (mean ± σ):  0.00% ± 0.00%
        █                                                        
  ▇▁▁▁▁▇█▁▁▁▁▁▁▇▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▇▁▇▁▁▁▁▁▁▁▁▇▁▁▁▁▁▁▇ ▁
  512 ms           Histogram: frequency by time          523 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.

• \(ijk\) and \(jik\) looping:

@benchmark matmul_by_loop!($A, $B, $C, "ijk")

BenchmarkTools.Trial: 21 samples with 1 evaluation per sample.
 Range (min … max):  238.764 ms … 284.262 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     241.943 ms               ┊ GC (median):    0.00%
 Time  (mean ± σ):   245.225 ms ±   9.869 ms  ┊ GC (mean ± σ):  0.00% ± 0.00%
  ▄▄█     ▁                                                      
  ███▁▆▁▆▆█▆▁▆▁▁▆▆▁▁▁▆▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▆ ▁
  239 ms           Histogram: frequency by time          284 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.

@benchmark matmul_by_loop!($A, $B, $C, "ijk")

BenchmarkTools.Trial: 21 samples with 1 evaluation per sample.
 Range (min … max):  238.929 ms … 251.045 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     243.993 ms               ┊ GC (median):    0.00%
 Time  (mean ± σ):   243.646 ms ±   3.958 ms  ┊ GC (mean ± σ):  0.00% ± 0.00%
    ▃█                              ▃                       ▃    
  ▇▁██▁▁▁▇▇▁▇▁▁▁▁▁▁▁▁▁▁▁▁▁▇▇▁▇▁▇▇▁▁▁█▁▇▁▇▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█▁▇ ▁
  239 ms           Histogram: frequency by time          251 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.

• Question: Can our loop beat BLAS? Julia wraps BLAS library for matrix multiplication. We see BLAS library wins hands down (multi-threading, Strassen algorithm, higher level-3 fraction by block outer product).

@benchmark mul!($C, $A, $B)

BenchmarkTools.Trial: 1766 samples with 1 evaluation per sample.
 Range (min … max):  2.549 ms …  11.043 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     2.691 ms               ┊ GC (median):    0.00%
 Time  (mean ± σ):   2.830 ms ± 481.713 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
  ▅▇██▆▅▄▄▄▄▃▃▂ ▁                                             ▁
  ███████████████▇▆▅▆▅▁▇▆▅▄▅▅▅▅▅▆▆▅▆▅▁▅▄▅▅▅▆▅▅▅▅▅▅▄▅▅▅▄▄▅▆▁▅▅ █
  2.55 ms      Histogram: log(frequency) by time      4.85 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.

# direct call of BLAS wrapper function
@benchmark LinearAlgebra.BLAS.gemm!('N', 'N', 1.0, $A, $B, 0.0, $C)

BenchmarkTools.Trial: 1778 samples with 1 evaluation per sample.
 Range (min … max):  2.536 ms …  12.463 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     2.689 ms               ┊ GC (median):    0.00%
 Time  (mean ± σ):   2.812 ms ± 438.347 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
  ▃▆██▇▆▄▅▄▄▃▃▂▂▁▁▁                                           ▁
  ██████████████████▅▇▅▆▅▅▅▇▅▄▅▅▁▅▅▅▅▄▅▇▅▅▅▅▄▅▅▆▄▅▄▅▅▄▄▅▅▁▁▅▅ █
  2.54 ms      Histogram: log(frequency) by time      4.65 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.

Question (again): Can our loop beat BLAS?

Exercise: Annotate the loop in matmul_by_loop! by @turbo and @tturbo (multi-threading) and benchmark again.

6 BLAS in R

• Tip for R users. Standard R distribution from CRAN uses a very out-dated BLAS/LAPACK library.

using RCall

R"""
sessionInfo()
"""

RObject{VecSxp}
R version 4.4.2 (2024-10-31)
Platform: aarch64-apple-darwin20
Running under: macOS Sequoia 15.4

Matrix products: default
BLAS:   /System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libBLAS.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.0

locale:
[1] C

time zone: America/Los_Angeles
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
[1] compiler_4.4.2

R"""
library(dplyr)
library(bench)
A <- $A
B <- $B
bench::mark(A %*% B) %>%
  print(width = Inf)
""";

┌ Warning: RCall.jl: 
│ Attaching package: 'dplyr'
│ 
│ The following objects are masked from 'package:stats':
│ 
│     filter, lag
│ 
│ The following objects are masked from 'package:base':
│ 
│     intersect, setdiff, setequal, union
│ 
└ @ RCall ~/.julia/packages/RCall/0ggIQ/src/io.jl:172

# A tibble: 1 x 13
  expression      min   median `itr/sec` mem_alloc `gc/sec` n_itr  n_gc
  <bch:expr> <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl> <int> <dbl>
1 A %*% B       127ms    136ms      6.77    30.5MB     6.77     4     4
  total_time result                memory             time          
    <bch:tm> <list>                <list>             <list>        
1      591ms <dbl [2,000 x 2,000]> <Rprofmem [1 x 3]> <bench_tm [4]>
  gc              
  <list>          
1 <tibble [4 x 3]>

┌ Warning: RCall.jl: Warning: Some expressions had a GC in every iteration; so filtering is disabled.
└ @ RCall ~/.julia/packages/RCall/0ggIQ/src/io.jl:172

• Re-build R from source using OpenBLAS or MKL will immediately boost linear algebra performance in R. Google build R using MKL to get started. Similarly we can build Julia using MKL.

• Matlab uses MKL. Usually it’s very hard to beat Matlab in terms of linear algebra.

using MATLAB

mat"""
f = @() $A * $B;
timeit(f)
"""

7 Avoid memory allocation: some examples

7.1 Transposing matrix is an expensive memory operation

In R, the command

t(A) %*% x

will first transpose A then perform matrix multiplication, causing unnecessary memory allocation

using Random, LinearAlgebra, BenchmarkTools
Random.seed!(123)

n = 1000
A = rand(n, n)
x = rand(n);

R"""
A <- $A
x <- $x
bench::mark(t(A) %*% x) %>%
  print(width = Inf)
""";

# A tibble: 1 x 13
  expression      min   median `itr/sec` mem_alloc `gc/sec` n_itr  n_gc
  <bch:expr> <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl> <int> <dbl>
1 t(A) %*% x   2.07ms   2.38ms      403.    7.64MB     80.1   146    29
  total_time result            memory             time            
    <bch:tm> <list>            <list>             <list>          
1      362ms <dbl [1,000 x 1]> <Rprofmem [3 x 3]> <bench_tm [175]>
  gc                
  <list>            
1 <tibble [175 x 3]>

Julia is avoids transposing matrix whenever possible. The Transpose type only creates a view of the transpose of matrix data.

typeof(transpose(A))

Transpose{Float64, Matrix{Float64}}

fieldnames(typeof(transpose(A)))

(:parent,)

# same data in tranpose(A) and original matrix A
pointer(transpose(A).parent), pointer(A)

(Ptr{Float64} @0x0000000130f48000, Ptr{Float64} @0x0000000130f48000)

# dispatch to BLAS
# does *not* actually transpose the matrix
@benchmark transpose($A) * $x

BenchmarkTools.Trial: 10000 samples with 1 evaluation per sample.
 Range (min … max):  22.583 μs … 250.292 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     23.375 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   24.822 μs ±   8.099 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
  █▇█▆▁ ▁▂▃▂▁                                                  ▂
  █████████████▆███▇▅▅▅▅▅▅▆▄▅▃▄▃▅▄▄▃▁▃▃▃▁▃▃▁▃▁▃▄▁▄▄▆▅▇██▇▆▆▄▅▅ █
  22.6 μs       Histogram: log(frequency) by time        55 μs <
 Memory estimate: 7.94 KiB, allocs estimate: 3.

# pre-allocate result
out = zeros(size(A, 2))
@benchmark mul!($out, transpose($A), $x)

BenchmarkTools.Trial: 10000 samples with 1 evaluation per sample.
 Range (min … max):  22.708 μs … 687.292 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     22.958 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   24.217 μs ±  10.551 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
  █▁    ▁▁▂▂                                                   ▁
  ██▇▇▆▇█████▇▇▅▆▇▇▆▆▆▆▅▄▄▅▅▄▄▅▄▄▄▁▄▁▄▄▃▁▄▁▁▃▄▄▁▃▁▃▁▃▁▁▃▄▄▆▇▇▆ █
  22.7 μs       Histogram: log(frequency) by time      50.8 μs <
 Memory estimate: 0 bytes, allocs estimate: 0.

# or call BLAS wrapper directly
@benchmark LinearAlgebra.BLAS.gemv!('T', 1.0, $A, $x, 0.0, $out)

BenchmarkTools.Trial: 10000 samples with 1 evaluation per sample.
 Range (min … max):  22.666 μs …  1.198 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     23.083 μs              ┊ GC (median):    0.00%
 Time  (mean ± σ):   24.330 μs ± 14.716 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
  █▅     ▁▁▁▁                                                 ▁
  ███▇▇▇▇████▇▆▆▇▇▇▇▅▆▅▅▅▄▄▄▅▄▄▃▃▄▁▄▅▄▁▃▁▃▁▃▄▄▄▅▃▃▁▁▁▁▁▄▅▇▆▆▅ █
  22.7 μs      Histogram: log(frequency) by time      51.6 μs <
 Memory estimate: 0 bytes, allocs estimate: 0.

7.2 Broadcast (dot operation) fuses loops and avoids memory allocation

Broadcasting in Julia achieves vectorized code without creating intermediate arrays.

Suppose we want to calculate elementsize maximum of absolute values of two large arrays. In R or Matlab, the command

max(abs(X), abs(Y))

will create two intermediate arrays and then one result array.

using RCall

Random.seed!(123)
X, Y = rand(1000, 1000), rand(1000, 1000)

R"""
library(dplyr)
library(bench)
bench::mark(max(abs($X), abs($Y))) %>%
  print(width = Inf)
""";

# A tibble: 1 x 13
  expression                           min   median `itr/sec` mem_alloc `gc/sec`
  <bch:expr>                      <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl>
1 max(abs(`#JL`$X), abs(`#JL`$Y))   3.44ms    3.8ms      250.    15.3MB     250.
  n_itr  n_gc total_time result    memory             time            
  <int> <dbl>   <bch:tm> <list>    <list>             <list>          
1    54    54      216ms <dbl [1]> <Rprofmem [2 x 3]> <bench_tm [108]>
  gc                
  <list>            
1 <tibble [108 x 3]>

In Julia, dot operations are fused so no intermediate arrays are created.

# no intermediate arrays created, only result array created
@benchmark max.(abs.($X), abs.($Y))

BenchmarkTools.Trial: 5140 samples with 1 evaluation per sample.
 Range (min … max):  208.917 μs …   9.703 ms  ┊ GC (min … max):  0.00% … 92.18%
 Time  (median):     723.541 μs               ┊ GC (median):     0.00%
 Time  (mean ± σ):   971.030 μs ± 615.106 μs  ┊ GC (mean ± σ):  26.78% ± 25.02%
   ▃▃      ▄▇█▇▅▄▃▂▁                ▁▁▁▂▂▂▂▂▂▁▁ ▁▁              ▂
  ▅███▇▃▁▁▅█████████▆▅▄▅▅▅▄▅▅▆▄▆▆▆██████████████████████▇██████ █
  209 μs        Histogram: log(frequency) by time       2.82 ms <
 Memory estimate: 7.63 MiB, allocs estimate: 3.

Pre-allocating result array gets rid of memory allocation at all.

# no memory allocation at all!
Z = zeros(size(X)) # zero matrix of same size as X
@benchmark $Z .= max.(abs.($X), abs.($Y)) # .= (vs =) is important!

BenchmarkTools.Trial: 10000 samples with 1 evaluation per sample.
 Range (min … max):  240.125 μs … 344.250 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     244.750 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   247.703 μs ±   8.097 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
     ▃▇█▆▄▅▅▅▄▃▂▂▂▂▂▁ ▁                                         ▂
  ▃▄▅█████████████████████▇█▇▇▇██▇▇▆▆▆▆▆▆▆▇▅▆▆▆▇▆▅▆▆▆▆▆▅▅▅▅▆▅▅▆ █
  240 μs        Histogram: log(frequency) by time        287 μs <
 Memory estimate: 0 bytes, allocs estimate: 0.

Exercise: Annotate the broadcasting by @turbo and @tturbo (multi-threading) and benchmark again.

7.3 Views

View avoids creating extra copy of matrix data.

Random.seed!(123)
A = randn(1000, 1000)

# sum entries in a sub-matrix
@benchmark sum($A[1:2:500, 1:2:500])

BenchmarkTools.Trial: 10000 samples with 1 evaluation per sample.
 Range (min … max):  31.625 μs …  12.441 ms  ┊ GC (min … max):  0.00% … 99.15%
 Time  (median):     59.500 μs               ┊ GC (median):     0.00%
 Time  (mean ± σ):   83.975 μs ± 286.746 μs  ┊ GC (mean ± σ):  28.74% ±  9.85%
   █                                                            
  ▃█▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▁▁▁▁▁▁▂▁▂▂ ▂
  31.6 μs         Histogram: frequency by time         1.23 ms <
 Memory estimate: 488.45 KiB, allocs estimate: 3.

# view avoids creating a separate sub-matrix
@benchmark sum(@view $A[1:2:500, 1:2:500])

BenchmarkTools.Trial: 10000 samples with 1 evaluation per sample.
 Range (min … max):  63.708 μs … 111.042 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     64.041 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   64.664 μs ±   2.378 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
  ▅█▆▃▁▁           ▁▁▁                                         ▁
  ███████▇▇▇▆▇▆▅▅▄▅█████▇██▇▆▅▆▆▅▅▅▅▄▅▆▆▇▇▆▇▇▆▆▆▅▅▆▄▅▃▁▅▅▅▆▆▆▆ █
  63.7 μs       Histogram: log(frequency) by time      75.9 μs <
 Memory estimate: 0 bytes, allocs estimate: 0.

The @views macro, which can be useful in some operations.

@benchmark @views sum($A[1:2:500, 1:2:500])

BenchmarkTools.Trial: 10000 samples with 1 evaluation per sample.
 Range (min … max):  63.709 μs … 111.750 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     64.000 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   64.774 μs ±   2.770 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
  ██▄▂▂▁       ▂▃ ▁▁                                           ▂
  ████████▇▇▆▅▅██████▇▇▆▅▅▅▅▅▁▆▇▆▇▇▇▇▇▆▅▆▆▄▄▅▄▆▆▅▇▇▅▆▅▅▅▄▄▄▄▅▆ █
  63.7 μs       Histogram: log(frequency) by time      79.2 μs <
 Memory estimate: 0 bytes, allocs estimate: 0.